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Sovereign Default Triggered by
Inability to Repay Debt
Michinao Okachi
Discussion Paper No. 2019-E-10
NOTE: IMES Discussion Paper Series is circulated in
order to stimulate discussion and comments. The
views expressed in Discussion Paper Series are
those of authors and do not necessarily reflect
those of the Bank of Japan or the Institute for
Monetary and Economic Studies.
IMES Discussion Paper Series 2019-E-10
July 2019
Sovereign Default Triggered by
Inability to Repay Debt
Michinao Okachi*
Abstract
The Greek sovereign default episode in 2012 was characterized by its high debt-to-
GDP ratio and the severe economic contraction following the default. Conventional
strategic default models designed to analyze a government's incentive to default
often fail to replicate these characteristics. To address this issue, we provide a
dynamic stochastic general equilibrium (DSGE) model where a sovereign default is
triggered by the government's inability to repay its debt. We show that the inability-
to-repay model replicates the empirical features observed in Greece, while the
conventional strategic default model calibrated to the Greek economy does not.
Keywords: Sovereign Default; Dynamic Stochastic General Equilibrium; Inability
to Repay Debt; Strategic Decision to Default; Fiscal Limit; Laffer Curve
JEL classification: E32, E44, F34, H63
* Economist, Institute for Monetary and Economic Studies, Bank of Japan (currently,
Department of Economics, Graduate School of Economics and Management, Tohoku University,
E-mail: [email protected])
The author thanks Junko Koeda, Ichiro Muto, Ko Nakayama, Taisuke Nakata, Eiji Okano,
Tatsuyoshi Okimoto, Tatsushi Okuda, Shigenori Shiratsuka, Nao Sudo, Shingo Watanabe,
Tomohiro Tsuruga, Chung Tran, Francesco Zanetti, participants at CEF 2018, the JEA 2017
Spring Meeting and 2018 Spring Meeting, the 4th Hitotsubashi Summer Institute, and seminar
presentations at Aoyama Gakuin University, Australian National University, Chuo University,
Hosei University, Kobe University, and Osaka University. The views expressed in this paper are
those of the author and do not necessarily reflect the official views of the Bank of Japan.
1 Introduction
The Greek sovereign default episode in 2012 was extraordinary in two respects.1 First,
the size of the public debt was the highest in the history of default, reaching 199.2 billion
euros at the outset of its default. The debt-to-GDP ratio was over 150 percent when
it defaulted, approximately twice as high as the average of sovereign defaults in other
countries that took place from 1982 to 2013, as shown in Figure 1 (a). Second, the Greek
default had a huge impact on its domestic economy in terms of severity and length of the
economic contraction after the default. GDP dropped by more than 20 percent, as shown
in Figure 1 (b). In addition, the economic contraction in Greece continued for more than
3 years. This is in contrast with what most defaulting countries witnessed in previous
episodes. As shown in Figure 1, on average, GDP in defaulting countries declined by
about 3 percent when the government defaulted and recovered to the pre-default level at
a faster pace, typically within 3 years.
Figure 1: Development of Debt-to-GDP ratio and GDP around Default Period
(a) Debt-to-GDP Ratio (b) GDP
3 2 1 0 1 2 3Year
0
50
100
150
200
Per
cent
AverageOne Std Deviation BandGreece (2012 episode)Argentina (2001 episode)
3 2 1 0 1 2 3Year
25
20
15
10
5
0
5
10
15
20
Per
cent
Notes 1: The horizontal axis represents the number of years after the onset of default.
2: The vertical axis in panel (b) represents the cumulative changes in GDP from 3 years
before default.
3: The numbers of default episodes used for computing debt-to-GDP ratio and GDP in the
�gure are 86 and 104, respectively. These episodes are taken from Schmitt-Grohe and
Uribe (2017).
Source: Historical Public Debt Database (IMF), and World Development Indicators (World Bank).
In the current paper, we argue that these empirical characteristics of the Greek de-
fault are not captured by conventional strategic default models in which the default is1In this paper, following the Standard and Poor�s de�nition, sovereign default is de�ned as the failure
of a government to meet a principal or interest payment on the due date. The duration of default isde�ned as the period that the country is not able to access bond markets after the default.
1
triggered by the government�s strategic decision. While some studies in the literature
apply conventional strategic default models to explain the European debt crisis including
Greece (e.g., D�Erasmo and Mendoza 2016, Salomao 2017, and Bocola and Dovis 2016),
they do not replicate the high debt-to-GDP ratio and the severe contraction observed in
the data.2 As seminal strategic model studies such as Aguiar and Gopinath (2006), Arel-
lano (2008), Yue (2010) and Mendoza and Yue (2012) state clearly that their models are
developed to analyze emerging countries, strategic default models are originally designed
to describe defaults in emerging countries in which the debt-to-GDP ratio tends to be
low and the economic contraction is less severe. Regarding the debt-to-GDP ratio, as Bi
(2012) and Paluszynski (2017) point out, the strategic default models cannot replicate the
high debt outstanding observed in the data. For example, the average debt-to-GDP ratio
in Aguiar and Gopinath (2006), Yue (2010), and Mendoza and Yue (2012) ranges from
10 percent to 23 percent, substantially lower than the level observed in Argentina (50
percent) or Greece (150 percent). To address this issue, Hatchondo and Martinez (2009),
Chatterjee and Eyigungor (2012) and Hatchondo, Martinez and Sosa-Padilla (2016) in-
troduce long-term bonds into sovereign default analysis to explain debt-to-GDP ratio.3
However, these studies are focused on explaining defaults in emerging economies and not
applied to Greece where debt-to-GDP ratio is substantially higher.
As for the severe and lengthy economic contraction, strategic default models cannot
replicate the data observed in Greece. This is partly because the models assume the
period of exclusion from �nancial markets after default is shorter than the data. For
example, Arellano (2008) and Yue (2010) set the average exclusion period to be less than
1 year so that the economy in the model recovers to the normal state at a rapid pace. In
the data, as reported in Dias and Richmond (2009), the exclusion period for defaulting
emerging countries ranges from 5.7 to 8.4 years. To address this issue, some recent studies
such as Benjamin and Wright (2009), Bi (2008) and Asonuma and Joo (2019a) analyze
the lengthy default duration as a result of prolonged debt restructuring negotiations.
Alternatively, Gordon and Gerron-Quintana (2018), Park (2017) and Asonuma and Joo
(2019b) emphasize the role of capital with which economic recovery can be delayed by
the weak investment after the default. However, none of these studies analyzed both
high debt-to-GDP and the severe economic contraction in Greece in a uni�ed modelling
framework. We argue that when the debt-to-GDP ratio is high and the adverse e¤ects
2Paluszynski (2017) and Bocola, Bornstein and Dovis (2019) point out that those models do notproperly describe the crisis.
3D�Erasmo (2011) emphasizes the role of the government�s reputation in explaining high debt-to-GDPratios.
2
of default are considerable as in the case of Greece, the government in strategic default
models is not willing to default because it is too costly.4 ;5
To address these issues, we propose a dynamic stochastic general equilibrium (DSGE)
model where the sovereign default is triggered by the government�s inability to repay
its debt, rather than the government�s strategic decision (Bi 2012, Bi and Leeper 2013,
Juessen, Linnermann and Schabert 2016). As Bi (2012) points out, this class of models
can describe high debt-to-GDP ratios observed in advanced countries including Greece.
Following Bi (2012), our inability-to-repay default model assumes that the government
defaults when the debt outstanding exceeds the sum of discounted maximum future �scal
surpluses determined by the shape of the La¤er curve. The government imposes a tax
on �nal goods, and issues new bonds in order to �nance its expenditure and repayment.
When debt-to-GDP ratio is low, the tax rate follows a linear rule. When debt-to-GDP
ratio is higher, the government tries to satisfy the intertemporal government budget con-
straint as much as possible by raising the rate nonlinearly. However, the government
cannot raise tax revenue beyond the �scal limit level. Once defaults, the government
introduces the austerity measure, namely an increase in the tax rate, to maximize tax
revenue. The high tax rate during this period leads to a severe economic contraction.
Similarly to Mendoza and Yue�s (2012) model, the economy su¤ers from the e¤ects of
�nancial autarky after the default as well as the adverse e¤ects arising from the high tax
rate, because domestic �rms cannot raise working capital necessary to import intermedi-
ate goods from abroad.
We compare this model with Mendoza and Yue (2012)�s strategic default model, both
of which are calibrated to the Greek economy. We conduct a quantitative simulation
that demonstrates the inability-to-repay default model closely captures the empirical
characteristics of the Greek episode, i.e., the high debt-to-GDP ratio at the time of
default and the severe contraction in GDP and its components following the default. In
contrast, the strategic default model cannot generate the default itself.
4In addition to these two issues, the discount factor in many strategic default models is set to a fairlylow value in order to replicate the default probability in the data. For example, the discount factor usedin Aguiar and Gopinath (2006), Yue (2010) and Mendoza and Yue (2012) ranges from 0.74 to 0.88, whichis substantially lower than the value widely used in existing studies. For example, Smets and Wouters(2003) use 0.99 as the discount factor.
5In fact, there are several anecdotes that support the view that Greek government tried hard to avoidthe default. For example, the government increased both value added and income tax rates, implementedcuts in salary for public employees and reduced public pension payments. Consequently, the tax revenue-to-GDP ratio surged from 38.9 percent in 2009 to 44.0 percent in 2011. Besides, the government hadbeen under strong external pressure from EU nations and the IMF to avoid the disorderly default whenreceiving emergency funding. These anecdotes are consistent with the empirical results of Levy-Yeyatiand Panizza (2011) that policymakers tend to defer default until the government�s insolvency becomeapparent.
3
This paper contributes to the literature in two aspects. First, to the best of our
knowledge, this is the �rst paper that develops an inability-to-repay type of sovereign de-
fault model that is calibrated in detail to the Greek economy.6 Second, we quantitatively
investigate which of the two default mechanisms can describe both the high debt-to-GDP
ratio and the severe economic contraction observed in Greek episode in a uni�ed frame-
work. We replicate both features with our calibrated inability-to-repay model. We also
show that the strategic default model such as Mendoza and Yue (2012) is not able to
explain Greek episode.7
This paper is organized as follows. In Section 2, we present both an inability-to-repay
default model and a strategic default model. In Section 3, we explain the calibration
strategy and solution methods. In Section 4, we conduct a quantitative analysis using
models calibrated to Greece and compare the implications of these two models. Finally,
we present concluding remarks in Section 5.
2 Model
In this paper, we construct two sovereign default models, each with its own distinct mech-
anism for triggering default. The common feature of the models is that both economies
are small open economies, consisting of four domestic agents: households, �nal goods
�rms, intermediate goods �rms and the government, and two foreign agents: foreign in-
vestors and foreign �rms. An overview of the model is given in Figure 2. The setting
of the private sector is borrowed from Mendoza and Yue (2012). Households consume
�nal goods and supply their labor to both �nal and intermediate goods �rms. Final
goods �rms produce �nal goods from labor and Armington-aggregated domestic and im-
ported intermediate goods with time varying TFP that follows the AR(1) process as:
At = �AAt�1 + "A;t, where "A � N(0; �2A). Imported intermediate goods consist of a
continuum of di¤erentiated intermediate goods that are aggregated using a Dixit-Stiglitz
aggregator. When importing the intermediate goods from abroad, �nal goods �rms need
to borrow working capital. This implies that when the government defaults, �nal goods
6In the literature, a range of research analyzes the interaction between sovereign default and nominaldevaluation (e.g., Na et al. 2018). In this paper, however, following the existing studies on the Greekdefault episode in 2012 (e.g., D�Erasmo and Mendoza 2016), we abstract such e¤ect on default. Thisis because Greece adopts euro as its currency, and its monetary policy is conducted by ECB which isliterally independent from the Greek government.
7Acharya and Rajan (2013) develops a model in which a sovereign default can be triggered by eitherthe inability to repay or by a strategic decision, and investigates the interaction between �scal stabilityand �nancial stability.
4
Figure 2: Model Overview
�rms lose access to the �nancial market and cannot import these intermediate goods.
The absence of imported intermediate goods leads to a loss in production e¢ ciency, and
results in a decline in �nal goods production. Intermediate goods �rms use only labor in
their production processes. See Appendix A for details of the settings regarding private
agents and the de�nition of the competitive equilibria under the two proposed models.
2.1 Government and Foreign Investors
At time t, the government repays its debt Bt issued in the previous period, spends govern-
ment expenditure Gt, collects tax revenue Tt, and issues new government bonds Bt+1 with
price qt. Government bonds are of one-period maturity, zero coupon and non-contingent,
taking both negative and positive values. A negative value for issuance of government
bonds indicates that the government�s net assets are positive and the government receives
interest income from foreign investors with the world risk-free interest rate rf . The gov-
ernment imposes a tax on production of �nal goods Yt. Thus, the total tax revenue Tt is
given by:
Tt = � tYt; (1)
5
where � t 2 [0; 1] is the tax rate. In normal times, the government issues new governmentbonds Bt+1 to meet the budget constraint:
qtBt+1 � Bt +Gt � Tt: (2)
If the government defaults, it does not repay its debt to foreign investors. Instead, inter-
national organizations (e.g. the IMF) bail out all of the debts and provide funding �Ftto the government to make up its budget de�cit. Note that at this state, the government
cannot issue new bonds by itself due to exclusion from the �nancial market. Exclusion
from the �nancial market continues for at least � periods. After � periods, the govern-
ment returns to the �nancial markets with the exogenous probability #, and with the
exogenously given debt outstanding Br.
Foreign investors are risk-neutral and behave as perfectly competitive agents, invest-
ing in both government bonds and working capital, taking all prices as given. If the
government defaults, these investors receive only a reduced amount of government bond
(1 � �)Bt from the international organizations, where � is the haircut rate. From the
expected zero pro�t condition of investment in government bonds, the bond is priced by:
qt =1
1 + rff(1� P et ) + P et (1� �)g; (3)
where P et is the probability of default in the next period.
2.2 Default Scheme
In this section, we describe separately the two distinct default mechanisms, inability-to-
repay debt and strategic decision to default.
2.2.1 Inability-to-Repay Default
Tax Rate Rule Under the inability-to-repay default model, the tax rate rule consists
of three parts.
First, when the debt outstanding is su¢ ciently low, similar to Davig, Leeper and
Walker (2010) and Bi (2012), the government increases its tax rate in proportion to the
debt outstanding:
� lrt = � + �(Bt �B); (4)
6
where � and B are the steady-state tax rate and debt outstanding that are exogenously
given, and �(� 0) is a tax rate adjustment parameter.Second, when the debt outstanding reaches a threshold value above which the govern-
ment cannot meet the intertemporal budget constraint, it adjusts the tax rate to satisfy
the constraint.8 The renewed intertemporal budget constraint is derived as:
Bt �nYj=0
qt+jBt+n+1 +
nXi=0
iYj=0
qt+j�1(T (� t+i; At+i)�G(gt+i)); (5)
where qt�1 = 1. Following Gali, López-Salido and Vallés (2007), gt is the government
expenditure expressed as deviations from the steady state, standardized by GDP, and
following the AR(1) process, i.e., gt = �ggt�1 + "g;t, where "g;t � N(0; �2g). The transver-
sality condition to ensure that the debt outstanding is not accumulated faster than the
discount rate is:
limn!1
nYj=0
qt+jBt+n+1 = 0: (6)
The intertemporal budget constraint with the transversality condition is derived as:
Bt �1Xi=0
iYj=0
qt+j�1(T (� t+i; At+i)�G(gt+i)): (7)
This constraint indicates that the current debt obligation has to be smaller than the sum
of the primary surplus over the entire future periods discounted by the price of government
bonds. The government tries to satisfy this constraint by changing the tax rate � t that
a¤ects the tax revenue Tt while the debt outstanding Bt is assumed to be predetermined
and government expenditure Gt is assumed to be exogenous. If the government maintains
the current tax rate � t in the future, the sum of the expected �scal surplus evaluated by
the price of government bonds St (hereafter, expected surplus) is:
St = S(� t; At; gt) = Et1Xi=0
iYj=0
qt+j�1(T (� t; At+i)�G(gt+i)): (8)
If the tax rate determined by the linear tax rate rule (4) does not satisfy the condition
Bt � S(� t; At; gt), i.e., the amount of debt outstanding Bt must be less than or equal to
8More precisely, the government sets the sequence of a speci�c tax rate that takes the same valuefrom the current period to the in�nite future, such that it satis�es the intertemporal government budgetconstraint.
7
the expected surplus, the government needs to deviate from the linear rule and adjust
the tax rate to satisfy the condition, Bt = S(� t; At; gt). Thus, the tax rate in this case,
� est , is:
� est = S�1(Bt; At; gt): (9)
The threshold of the debt outstanding Bt at which the debt outstanding exceeds the
expected surplus under the linear rule is denoted by Zint(At; gt).
Third, we assume that the tax rate is bounded from above. The government is not
able to set a tax rate above some point at which the direct increase in tax revenue from
the tax rate increase is fully o¤set by the decrease in tax revenue that arises from the
drop in �nal goods due to the increase in the tax rate. In other words, in this case, the
economy is at the top of the La¤er curve. Moreover, following Bi, Shen and Yang (2014),
we assume that the government faces political disturbance % 2 [0; 1] that prevents thegovernment from setting the tax rate at the top of the curve. This political disturbance is
interpreted as policymakers�lacking the power or willingness to impose a severe austerity
policy because such policy is often accompanied by a heavy burden for taxpayers. Thus,
the maximum tax rate that the government can set, �mat , is:
�ma(At) = (1� %) ��argmax
� tT (� t; At)
�: (10)
The sum of expected maximum �scal surplus discounted by the price of government
bonds, denoted by Zmat , is:
Zma(At; gt) = Et1Xi=0
iYj=0
qt+j�1(T (�mat+i; At+i)�G(gt+i)): (11)
If the debt outstanding exceeds the expected maximum surplus Zma(At; gt), the govern-
ment does not increase the tax rate further, because of political disturbance, and declares
a default. The government�s tax rate rule is summarized as:
� t =
8>>>>>><>>>>>>:� lrt if Bt � Zint(At; gt);
� est if Zint(At; gt) < Bt � Zma(At; gt);
�mat if Bt > Zma(At; gt) .
(12)
8
Figure 3 illustrates this tax rate rule.9 ;10
Figure 3: Tax Rate Rule under Inability-to-Repay Default Model
Finally, in the default state, the government is forced by international organizations to
introduce the austerity measure and to set its tax rate so that it can collect the maximum
tax revenue without political disturbance, namely the rate that corresponds to the top
of the La¤er curve (� tot ).
Default Region In the inability-to-repay default model, the government defaults when
the debt outstanding exceeds the �scal limit. The �scal limit b�t is drawn from the
distribution of the sum of discounted maximum �scal surpluses denoted by B�:11
B�(At; gt) =
1Xi=0
iYj=0
qt+j�1(T (�mat+i; At+i)�G(gt+i)): (13)
Whether the economy is in the default state or not at the beginning of period t is
denoted by �t 2 f�n;t; �d;tg, where �n;t and �d;t represent non-default state and default9If Zint(At; gt) > Zma(At; gt), the tax rate rule only consists of two parts: � lrt and �
mat .
10As shown in Figure 3, the linear tax rule alone does not correspond to the default at expectedmaximum surplus, failing to explain the high debt to GDP upon default as observed in the data.
11As Ghosh et al. (2013) points out, a small shock to the current primary balance can lead to asubstantial di¤erence in the �scal limit. For example, if the interest rate is 1 percent, a 1 percentagepoint increase in the current �scal surplus relative to GDP pushes up the �scal limit relative to GDP by100 percentage points.
9
state respectively.12 The default region �I(Bt; �n;t; b�t ) that shapes the set of TFP and
government expenditure in which the debt outstanding exceeds the �scal limit in the next
period is de�ned as:
�I(Bt; �n;t; b�t ) = fAt 2 A, gt 2 G : Bt > b�t , �t = �n;tg: (14)
The probability of inability-to-repay default in the next period P I;et is the probability
that TFP and government expenditure shocks fall into the default region:
P I;e(At; gt; Bt+1; �n;t+1; b�t+1) =
Z Z�I(Bt+1;�n;t+1;b
�t+1)
fA(At+1; At)fg(gt+1; gt)dAt+1dgt+1;
(15)
where fA and f g are the transition probability function of TFP and government expen-
diture respectively. Note also that, foreign investors are assumed to price the government
bonds depending on the probability of inability-to-repay default, i.e., the probability of
default is set at P et = P I;et .
2.2.2 Strategic Default
In this subsection, we explain the alternative default mechanism; strategic decision to
default that is widely studied in the existing literature. In this alternative model, the
government chooses to default if and only if defaulting is the better option for the gov-
ernment.13
Tax Rate Rule Similarly to existing studies, we assume that the government sets its
tax rate following the linear rule (4).
Default Region The government makes the decision regarding its default so as to
maximize households� lifetime utility. Thus, the government�s optimization problem is
12Obviously, the transition to the default state can take place only from the non-default state (i.e.�t = �n;t).
13The implicit assumption commonly made in strategic default models is that there is an upper boundregarding the debt-to-GDP ratio in the economy, and the government decides to default or not underthis upper bound. This assumption rules out Ponzi schemes. For example, Mendoza and Yue (2012)assume the largest debt amount the government can repay with full commitment as output over risk-free interest rate. One key di¤erence between this upper bound and the �scal limit considered in theinability-to-repay model is that the former considers the use of lump-sum tax whereas the latter doesnot. Following the convention of the strategic default model, we assume in the analysis below that thegovernment exploits the lump-sum tax to meet the intertemporal government budget constraint whenstudying the strategic default model. Note also that when studying the inability-to-default model, weassume that the government is unable to use a lump-sum tax.
10
formulated as the maximization of its value function de�ned as:
V (At; gt; Bt; �n;t) = maxdt2f0;1g
f(1� dt)Vn(At; gt; Bt; �n;t) + dtVd(At; gt; �n;t)g; (16)
where dt represents the government�s default decision, taking 1 for default and 0 for non-
default, and Vn and Vd represent the government�s value functions corresponding to the
non-default state and default state respectively. They are de�ned as:
Vn(At; gt; Bt; �n;t) = u(Ct; Lt) + �Et[V (At+1; gt+1; Bt+1; �n;t+1)j(At; gt; Bt; �n;t)]; (17)
and
Vd(At; gt; �n;t) = u(Ct; Lt) + �f(1� #)Et[Vd(At+1; gt+1; �d;t+1)j(At; gt; �n;t)] (18)
+#Et[Vn(At+1; gt+1; Br; �n;t+1)j(At; gt; �n;t)]g;
where � 2 (0; 1) is the discount factor and # is an exogenously given recovery probabilitywith which the defaulted government reverts back to the non-default state.14 The default
region �S and the probability of default P S;e are de�ned as:
�S(Bt; �n;t) = fAt 2 A, gt 2 G : Vd(At; gt; �n;t) > Vn(At; gt; Bt; �n;t), �t = �n;tg; (19)
and
P S;e(At; gt; Bt+1; �n;t+1) =
Z Z�S(Bt+1;�n;t+1)
fA(At+1; At)fg(gt+1; gt)dAt+1dgt+1: (20)
The price of the government bond depends on the probability of strategic default as
P et = P S;et .
3 Calibration and Solution Method
For the purpose of the quantitative exercise focusing on the Greek default episode in
2012, we calibrate the two models to the Greek economy. In this section, we explain the
calibration strategy and solution method.
14In this model, following Mendoza and Yue (2012), we assume � = 0.
11
3.1 Calibration Strategy
Most parameters are calibrated to the data from 1999Q1 to 2016Q4 except some deep
parameters that are set to be consistent with existing studies.
Government Regarding the parameters related to the government�s bond price, the
discount factor � is set at 0.99 following Smets and Wouters (2003), which implies the
risk-free interest rate rf is 0.01. We set the haircut rate � to 0.05 so as to be consistent
with the actual bond spread, i.e., the quarterly long-term interest rate di¤erential of
Greek government bonds relative to the German bonds in 2012.15 The government debt
outstanding at the steady state B is set so that the debt-to-GDP ratio at steady state is
104 percent, which matches with the historical average in Greece from 1999 to 2008.16 ;17
As for the parameters related to �scal surplus/de�cit, we set the steady state govern-
ment expenditure-to-GDP G to 0:385, to match the data from 1999 to 2008.18 To close
the �scal surplus/de�cit at the steady state, the steady-state tax rate � is set so that the
tax revenues are equalized to G.
Private Agents The households�utility function is of the Greenwood-Hercowitz-Hu¤man
(1988) preference19 as u(ct; Lt) =
�Ct�
L1+�t1+�
�1���1
1�� , where � is the degree of risk aversion,
which we set to unity so that the function is the log-utility, and � is the inverse Frisch
elasticity of labor supply, which we set at 0.455, the standard value used in existing RBC
models.
As for �rms, the intermediate input share in �nal goods production �M is set to
0:39 to match the total intermediate consumption over gross �nal goods in Greece in
the early 2000s, taken from the OECD�s STAN Input-Output data. The labor input
share in �nal goods production �L is set at 0:42 so that the proportion of labor input
to total input minus intermediate input share (i.e., �L=(1 � �M)) is equivalent to 0.7,
15More precisely, the haircut rate is set by matching the model produced bond spread with theobserved spread. The ex-post haircut rate of Greek debt restructuring was about 60 percent (Zettelmeyer,Trebesch and Gulati 2013). However, the amount of defaulted debt was limited to the equivalent of about100 percent of its GDP. GDP-linked warrants were also granted to investors. Thus, the e¤ective haircutrate could be much lower than the ex-post haircut rate observed in the data.
16The reason we drop the data in 2008 is that this is 1 year ahead of the year when underreportingof the government�s debt outstanding and de�cit is revealed.
17The government debt outstanding is not measured on a net basis but on a gross basis because ofdata limitations. However, as the gap between gross and net debt was less than 10 percent of GDP inGreece, it does not have any signi�cant quantitative e¤ect on the result.
18Government expenditure is measured as its total expenses, including social bene�ts, and excludinginterest payments.
19Under the Greenwood-Hercowitz-Hu¤man preference, labor supply depends only on the wage rate.
12
which is often assumed in most literature. The Armington weight of domestic inputs
� is set at 0:63; to match the share of domestic intermediate goods-to-GDP, taken from
the OECD�s STAN Input-Output data. The substitution elasticity between domestic and
foreign intermediate goods is set at 0:9; following Evers (2015). Dixit-Stiglitz curvature
parameter � is set at 3; following Feenstra et al. (2018). The upper bound of imported
inputs with working capital � is set at 0.26 according to the World Bank data in 2005,
and time invariant TFP for intermediate goods production AI is set at 0.31, following
Mendoza and Yue (2012). The labor share in the production of intermediate goods is
set at 0.7.
Other Parameters Parameters associated with shocks are set using HP-�ltered cor-
responding time series. These parameters include the auto-regressive parameter and the
standard deviation of TFP shock, 0.600 and 0.015, and those of government expenditure
shock, 0.650 and 0.040 respectively. We set the period of exclusion from the �nancial mar-
ket after default � to 26 quarters,20 and the probability of recovery to the non-default
state # to 0.044 following Dias and Richmond (2009).
Tax Rate Rule under Inability-to-Repay Default Model To calibrate the para-
meters regarding the �scal rule we observe developments in the tax rate and the debt-
to-GDP ratio in Greece. Figure 4 (a) shows the historical relation between debt-to-GDP
ratios and tax rates.
The data can be roughly categorized into four sub-periods. First, before the debt-to-
GDP ratio hit around 100 percent, namely from 1984 to 1992, these two variables were
positively correlated. Second, from 1993 to 2008, the tax rate ranged between 30 percent
and 40 percent while the debt-to-GDP ratio remained at around 100 percent. Third, after
the onset of the global �nancial crisis in 2008, the debt-to-GDP ratio rose at quite a rapid
pace, while the pace of tax increase was moderate. Finally, the government defaulted in
2012, and both the debt-to-GDP ratio and the tax rate have been elevated since then.
In the model, the parameter of the linear tax rate rule is calibrated to the data from
1984 to 1992 using OLS estimation. The political disturbance % is set at 0.08, so that
the tax rate is consistent with the observed rate. In 2011, a year before the default,
the tax rate was about 8 percentage points lower than the post-default average during
2012-16, which implies that the government could have increased the tax rate by about
8 percentage points to increase tax revenue, if it had not been facing political concerns.
20This is because it took about 6.5 years for the Greek government to complete the �nal bailoutprogram funded by the European Stability Mechanism (ESM).
13
Table 1: Calibration to the Greek Economy
Parameter Value Target/Source�: Risk aversion 1 Log-utility�: Inverse Frisch elasticity of labor supply 0.455 Standard value�: Households�discount factor 0.99 Smets and Wouters (2003)�M : Intermediate input share in �nal goods production 0.39 OECD�L: Labor input share in �nal goods production 0.42 Standard value�: Armington weight of domestic inputs 0.63 OECD : Substitution elasticity across intermediate goods 0.9 Evers(2015)�: Substitution elasticity within intermediate goods 3 Feenstra et al. (2018)�: Imported goods with working capital 0.26 World BankAI : TFP for intermediate goods production 0.31 Mendoza and Yue (2012)rf : World risk-free interest rate 0.01 Smets and Wouters (2003) : Labor input share in intermediate goods production 0.7 Standard value�: Haircut rate 0.05 Bond premium in 2012�g: Persistence of gov�t expenditure shock 0.65 IMF�g: Standard deviation of gov�t expenditure shock 0.04 IMF�A: Persistence of TFP shock 0.6 OECD�A: Standard deviation of TFP shock 0.015 OECDG: Steady-state government expenditure 0.0163 IMF (G=GDP = 0:385)B: Steady-state government debt 0.0440 IMF (B=GDP = 1:04)� : Steady-state tax rate 0.385 Steady-state gov�t expenditure�: Elasticity of tax rate 0.02 OLS%: Political disturbance 0.08 IMF�: Period of exclusion 26 Duration of Bailout Program#: Probability of recovery 0.044 Dias and Richmond (2009)
Figure 4 (b) shows the calibrated tax rate rule under the inability-to-repay default
model. In the area where the debt-to-GDP ratio is below 100 percent and the tax rate
is below 39 percent, the government sets its tax rate according to the linear rule (line
marked with circles). In the area where the debt-to-GDP is higher than 100 percent
and the tax rate is above 39 percent, the line exceeds the expected surplus (solid curve)
derived from the equation (8). In this area, to sustain the high debt-to-GDP ratio, the
government deviates from the linear rule and sets the higher tax rate on the curve. Once
the tax rate reaches the maximum tax rate that is estimated to be 40.5 percent, the
government cannot contain any further increase in the debt-to-GDP ratio by raising the
tax rate, because of political pressures, and declares the default.
Table 1 summarizes all calibrated parameters for the Greek economy.21
21The explanation of calibrated parameter values for the Argentinean economy is provided in AppendixE.
14
Figure 4: Tax Rate Rule
(a) Greek Data (b) Calibrated Result
0.25 0.3 0.35 0.4 0.45 0.5 0.55Tax Rate
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Deb
t / G
DP
19841992
20092011
20122016
19932008
Notes 1: The level of GDP in the right panel (b) is the steady-state value.
2: ES represents the expected surplus.
3: "High A" and "Low A" represent 5 percent higher and lower than the steady-state TFP respectively.
Source: OECD
3.2 Solution Method
There are two steps to solving the inability-to-repay model. In the �rst step, we compute
the mapping from a given productivity level At and expenditure level gt to the expected
surplus and �scal limit,22 since the �scal limit is a function of realizations of future shocks
to these variables that are drawn by Monte-Carlo sampling. In the second step, based
on the mapping, we solve the model on the discrete state space (DSS). To obtain the
DSS, following Coleman (1991) and Davig (2004), we �nd a �xed point of the decision
rule for government bonds by using the monotone map method for each of three tax rate
cases, � lrt , �est and �mat respectively. Then, we select the sequences of the tax rate and
corresponding endogenous variables, following the tax rate rule in (12).
As for the solution of the strategic default model, we apply a two-loop algorithm.
The outside and inside loops iterate the price of government bonds and government value
functions in both non-default and default states respectively. A detailed explanation of
the computational procedure is given in Appendix B.
22Following Bi (2012), to compute the �scal limit, we assume that the future government bond ispriced under the assumption of a zero default probability, i.e., the price of government bonds is onlydiscounted by the risk-free interest rate.
15
4 Quantitative Analysis
In this section, we quantitatively investigate how each of the two proposed models �ts
with the Greek experience. As we explained in the previous section, the government
defaults when the current TFP and government expenditure fall into the default region.
In what follows, we study the default regions under the two proposed models and
investigate which of the two models can provide more plausible explanation of the Greek
default episode in 2012. Then we examine the simulated path of the economy under
the two models and see whether they can capture the severe and long-lasting economic
contraction around the default.
4.1 Trigger of Default
To investigate the trigger of the default, we compute the default regions constructed by
the two models.
First, we compute the La¤er curve of the economy in the inability-to-repay default
model. Panel (a) in Figure 5 shows the La¤er curve under di¤erent values of TFP,
taking the tax rate on the horizontal axis and the tax revenue over steady-state GDP
on the vertical axis.23 The tax rate at the top of the La¤er curve is about 44.5 percent,
and the maximum tax rate is about 40.5 percent. The maximum tax revenue over the
steady-state GDP is about 40 percent. This means that the maximum �scal surplus is
about 1.5 percent, i.e., the government can potentially accumulate about 1.5 percent of
�scal surplus over GDP, provided that the steady-state government expenditure-to-GDP
is maintained at 38.5 percent. When TFP is 5 percentage points higher (lower), the
tax revenue relative to GDP is about 5 percentage points higher (lower) than the steady
state. Panel (b) illustrates the probability of inability-to-repay default under di¤erent
values of current TFP, where we suppose that government expenditure is �xed at the
steady state. If the current TFP is at the steady state (bold line) and the debt-to-GDP
ratio is around 100 percent, then the probability of default is around 10 percent. The
probability increases with debt-to-GDP ratio and reaches 50 percent when the debt-to-
GDP ratio reaches about 160 percent.
Next, we show the value functions for the strategic default model. Panel (c) shows
the di¤erence between the two value functions, i.e., the value function if the government
chooses not to default and if it chooses to default, under di¤erent values of the current
23The steady-state GDP is computed, by feeding into the model the steady-state values of TFP andgovernment expenditure.
16
Figure 5: Comparison of Two Sovereign Default Models (Greece)
(b) Probability of(a) La¤er Curve (LC) Inability-to-Repay Default
0.3 0.35 0.4 0.45 0.5Tax Rate
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
Tax
Rev
enue
/ G
DP
High ASteady ALow A
MaxFiscalSurplus
G / GDP
TauMax Top
of LC
0 0.5 1 1.5 2 2.5 3Debt / GDP
0
0.2
0.4
0.6
0.8
1
Low ASteady AHigh A
(c) Value of Strategic Default (d) Default Region
0 0.5 1 1.5 2 2.5 3Debt / GDP
0
0.2
0.4
0.6
0.8
1
Vn
Vd
High ASteady ALow A
7.5
5.6
3.7
1.9
0
1.9
3.8
5.6
7.5
TFP
0 0.5 1 1.5 2 2.5 3Debt / GDP
Nondef ault
Def aultedPoint
Incapability toRepay
Notes 1: Debt-to-GDP is measured relative to the steady-state GDP.
2: TFP in panel (d) shows the deviation from the steady state.
3: Government expenditure is �xed at the steady state.
TFP. The di¤erence is positive over the plausible range of debt-to-GDP ratio below 300
percent. This suggests that the government has no incentive to choose to default.
For comparison, panel (d) illustrates the default regions with various values of TFP
given that government expenditure is �xed at the steady state. The default region of the
inability-to-repay default is located in the area where the debt-to-GDP ratio is higher than
150 percent, while the region of strategic default does not appear in this panel as implied
in panel (c). In other words, over the range considered in panel (d), the government does
not choose to default. The dotted point in the panel represents the pair of observations
of debt-to-GDP ratio and TFP that was observed at the onset of the default in Greece
in 2012. Overall, these pictures suggest that the Greek default was likely attributable
17
to the government�s inability to repay its debts rather than the government�s strategic
decision.24
4.2 Simulation Analysis
In this section we examine the quantitative performance of the inability-to-repay default
model in explaining the severe economic contraction after the default in Greece. We
set the initial debt-to-GDP ratio to 80 percent, and the initial values of both TFP and
government expenditure at the steady-state values. Then we generate stochastic TFP
and government expenditure shocks 5,000 times over 500 quarters, and compute the
equilibrium time path of endogenous variables.
Table 2: Summary Statistics
Average Before Default Average After Default2009Q1-2011Q4 2012Q1-2015Q1
Unit Data Model Data ModelDebt-to-GDP Percent 141.1 148.1 167.6 170.9Probability of Default Percent 34.5 42.1 � �Bond Spread % Points 7.2 7.0 � �Tax Rate Percent 36.4 39.8 42.6 44.3
Table 3: Cumulative Changes around the Default
Economic Impact by DefaultUnit Data Model
GDP Percent -21.0 -13.8Consumption Percent -20.1 -21.6Imports Percent -13.2 -31.0
Note: Cumulative changes represent the percent change of the variables from 3 years before the default
to 3 years after the default.
Table 2 reports the summary statistics of the key variables in the data and simulation
before and after the default. This suggests that the inability-to-repay default model
successfully replicates these variables.25
Regarding other endogenous economic variables, Table 3 shows the cumulative changes
in GDP and its components from 3 years before the default until 3 years after the default.24In Appendix C, we conduct robustness analysis for di¤erent values of government expenditure and
con�rm that the results are not substantially altered.25The bond spread in the data is the 10-year Greek government bond yield against the 10-year German
government bond yield on a quarterly basis.
18
This suggests that the model can replicate the signi�cant drop in GDP, consumption, and
imports around the default.
Figure 6: Key Endogenous Variables (Simulation vs Data)
(a) Debt-to-GDP ratio (b) Government Bond Spread
3 2 1 0 1 2 3Year
110
120
130
140
150
160
170
180
190
Perc
ent
DataSimSim,AltSim,Band
3 2 1 0Year
0
5
10
15
20
25
Perc
enta
ge P
oint
(c) Tax Rate (d) GDP
3 2 1 0 1 2 3Year
32
34
36
38
40
42
44
46
48
50
Perc
ent
3 2 1 0 1 2 3Year
25
20
15
10
5
0
5
Perc
ent
(e) Consumption (f) Imports
3 2 1 0 1 2 3Year
30
25
20
15
10
5
0
5
10
Perc
ent
3 2 1 0 1 2 3Year
40
35
30
25
20
15
10
5
0
5
Perc
ent
Notes 1: Year 0 is set at 2012Q1 for the data.
2: 2Y-Alt simulation represents simulation results with 2 years of exclusion from the
�nancial market.
Figure 6 illustrates the developments in the key variables in both data and simulation
19
before and after the default. Panel (a) shows the mean of the baseline simulation path of
debt-to-GDP ratio for defaulted samples. The simulated path (bold line) closely tracks
the realized path of the debt-to-GDP ratio within the one standard error band. Panel
(b) shows the mean of the simulation path of the spread. It illustrates that the nonlinear
increase in spread is somewhat captured by the simulation path, although some periods of
deviation from the error band are observed. Given the market turbulence in this period,
the deviation could be explained by other factors such as the e¤ect of overall uncertainty
regarding the euro area sovereign debt problem. Panels (d), (e), and (f) describe the
cumulative changes in GDP, consumption, and imports from 3 years before the default.
The model closely replicates the data as the deviations fall mostly in the one standard
error band. 26 ;27
Overall, the model generally accommodates features of the highly nonlinear dynamics
of key economic variables around the default.
4.3 Discussion: Argentinean Default in 2001
We have so far focused on the case of the Greek default throughout this paper. Because a
good number of recent studies of the sovereign default analyze the Argentinean default in
2001, however, it would be worthwhile to apply both the inability-to-repay default model
and the strategic default model to the Argentinean economy and highlight the di¤erences
between the two models.28
Essentially, the Argentinean default is attributable to the government�s strategic deci-
sion rather than its inability to repay its debt. The Argentinean default is not considered
as having been triggered by the government�s inability to repay its debt because, given
that economic contraction following the default was moderate, the government was con-
sidered to have been less reluctant to choose default. Panel (a) in Figure 7 shows the
La¤er curve in Argentina and the maximum �scal surplus. The tax revenue over GDP
that the government can achieve when the tax rate is set at the top of the La¤er curve
is about 32 percent, which is about 10 percentage points larger than the steady-state
government expenditure. Following Bi, Shen and Yang (2014) we set % to 0.33.29 As a
26In this analysis, we assumed that the period of exclusion from �nancial markets is at least 26quarters. This is relatively long when compared with existing studies.
27To verify the robustness of the results, we also present the mean of simulated path in which thedefault state continues for only 8 quarters, the standard assumption used in the literature. We �nd thatthe overall picture is little altered by this change in the setting.
28We explain the calibration strategy and results of the Argentinean economy in Appendix E.29Bi, Shen and Yang (2014) applies an inability-to-repay type model to Argentinean economy and
subtract 33 percent of the tax rate in all future periods by referring to the Argentinean political risk
20
Figure 7: Comparison of Two Sovereign Default Models (Argentina)
(a) La¤er Curve (LC) (b) Value of Strategic Default
0.2 0.3 0.4 0.5 0.6Tax Rate
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Tax
Rev
enue
/ G
DP
High ASteady ALow A
Top ofLC
Max FiscalSurplus
G / GDP
0 0.2 0.4 0.6 0.8 1Debt / GDP
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
Vn
Vd
High ASteady ALow A
(c) Default Region
7.5
5.6
3.7
1.9
0
1.9
3.8
5.6
7.5
TFP
0 0.2 0.4 0.6 0.8 1Debt / GDP
Nondefault
DefaultedPoint
StrategicDefaul t
Notes 1: Debt-to-GDP is measured relative to the steady-state GDP.
2: TFP in panel (c) shows the deviation from the steady state.
3: Government expenditure is �xed at the steady state.
result, the �scal surplus would be about 5 percent at the steady state and the potential
�scal limit is about 250 percent of GDP, which is substantially higher than the actual
debt-to-GDP ratio.
Regarding the strategic default model, we assume that the tax rate in the default
state is 5 percentage points lower than that in the non-default steady state, based on
the fact that the historical average of the tax rate in Argentina during 2002-03 was
5 percentage points lower than that of 1998-2002 in Argentina. Panel (b) shows the
di¤erence of value functions constructed by the strategic default model, i.e., non-default
value function minus the default value function. Given the steady-state value of TFP,
the value function at the default state exceeds the value function at the non-default state
factor taken from the International Country Risk Guide�s (ICRG�s) index in order to replicate the low�scal limit in emerging countries such as Argentina.
21
for debt-to-GDP ratio by more than 70 percent. In other words, this value is considered
as the threshold at which strategic default is triggered.
Panel (c) illustrates the default region for various values of TFP and debt-to-GDP ra-
tio. Again, we assume that government expenditure is at the steady state. The inability-
to-repay default region does not appear in this picture where debt-to-GDP ratio ranging
from 0 to 100 percent is considered. The actual TFP and debt-to-GDP ratio at the time
the government defaulted in 2001 falls into the strategic default region in the panel.
The key di¤erence from the Greek case is that Argentina had more room to accumulate
future �scal surplus than Greece, which in turn made inability-to-repay default less likely.
Besides, choosing to default was considered less attractive to the Greek government than
the Argentinean government. This is because, given the low tax rate after default in
Argentinean data at the time of its default, economic contraction was relatively mild,
and the government could improve households�welfare by choosing to default and relieve
itself of having to continuously make interest payments.
5 Conclusion
We propose a DSGE model in which a sovereign default is triggered by a government�s
inability to repay its debt because the government has limited capacity to increase its �scal
surplus. We compare this model with the strategic default model in which the government
chooses to default to avoid the burden of debt. We conduct a quantitative analysis that
shows the Greek default in 2012 was attributable to the government�s inability rather
than to its strategic decision, this being consistent with the view that, fearing default
would lead to a large economic contraction, the Greek government had no incentive to
choose to default. In addition, our simulation study shows that the model can replicate
the high debt-to-GDP ratio and the substantial drop in GDP and its components.
Admittedly, the economic environments considered in the current paper are quite sim-
ple, and there is potential room to add various elements in the economy not covered here
that have an important bearing upon a government�s decision to default. For example,
as Borensztein and Panizza (2009) mentioned, to bring the model closer to the data, it
may be bene�cial to consider the banking sector and the government�s strategic decisions
that depend on which sector is holding bonds. Further research along these lines is left
as our future research agenda.
22
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25
Appendix A. Private Agents and Competitive Equi-librium
A.1 Households
A representative household derives utility from consumption Ct and disutility from
labor Lt. The household provides its labor Lt to both �nal and intermediate goods �rms,
and receives labor income wtLt from these two types of �rms and pro�ts �ft and �mt
from each type of �rm respectively, where wt is the wage rate. The household solves the
following utility maximization problem:
maxCt;Lt
Et
" 1Xi=0
�iu(Ct+i; Lt+i)
#; (21)
subject to its budget constraint Ct � wtLt + �ft + �mt , where � 2 (0; 1) is the subjectivediscount factor, and the utility function u : R2+ ! R is continuous, twice di¤erential andsatis�es @u
@C> 0; @
2u@C2
< 0; @u@L< 0; @
2u@L2
< 0 and @2u@C2
@2u@L2
��
@2u@C@L
�2> 0.
A.2 Final Goods Firms
There is a continuum of �nal goods �rms, producing �nal goods from intermediate
goods Mt and labor inputs Lft .30 The production function follows Cobb-Douglas:
Yt = eAt�M�mdt ;m
�t
���M (Lft )�L ; (22)
where M�mdt ;m
�t
�is intermediate goods which are composed of domestic intermediate
inputs mdt and imported intermediate inputs m
�t . Parameters 0 < �M ; �L < 1 are the
share of intermediate goods and labor in �nal goods respectively, satisfying �M +�L < 1.
The technology of combining these two types of intermediate goods follows the CES
Armington aggregator:
M�mdt ;m
�t
�=
���mdt
� �1 + (1� �) (m�
t ) �1
� �1
; (23)
where � 2 [0; 1] is the weight of domestic intermediate goods in total intermediate goodscomposition and (> 0) is the elasticity of substitution across domestic and imported
30Following Mendoza and Yue (2012), we abstract capital input for simplicity.
26
intermediate goods. The pro�t maximization problem of �nal goods �rms is:
maxm�t ;m
dt ;L
ft
�ft = Yt(1� � t)� p�tm�t � pmt m
dt � wtL
ft ; (24)
subject to the production function (22) and the combining technology of domestic and
imported intermediate goods (23). As explained above, the government imposes a tax on
their �nal goods production.
Then, the �rst-order conditions are:
eAt�M�M�mdt ;m
�t
���M� �1 (1� �)(m�
t )�1 �Lft
��L(1� � t) = p�t ; (25)
eAt�M�M�mdt ;m
�t
���M� �1 �
�mdt
��1
�Lft
��L(1� � t) = pmt ; (26)
eAt�L�M�mdt ;m
�t
���M �Lft ��L�1 (1� � t) = wt: (27)
The technology of combining the continuum of di¤erentiated imported intermediate goods
m�j;t is the Dixit-Stiglitz aggregator:
m�t �
�Z 1
0
�m�j;t
� ��1� dj
� ���1
; (28)
where �(> 0) is the substitution elasticity among imported intermediate goods. If the
government repays its debt, �nal goods �rms can access the �nancial market to borrow
working capital at the risk-free interest rate rf from foreign investors in order to import
a fraction � 2 [0; 1] of intermediate goods from foreign �rms. The foreign investors have
enough resources to lend the working capital, and �nal goods �rms repay it within the
period t. Thus, the cost minimization problem with respect to the imported intermediate
goods is:
minm�jt
Z 1
�
p�j;tm�j;tdj + (1 + rf )
Z �
0
p�j;tm�j;tdj: (29)
Then, the �rst-order conditions with respect to m�j;t are as follows:
m�j;t=
(1 + rf )p�j;t
p�t
!�vm�t ; for j 2 [0; �]; (30)
m�j;t=
�p�j;tp�t
��vm�t ; for j 2 [�; 1]: (31)
27
The aggregate imported intermediate goods is provided by the CES index:
p�t =
�Z 1
�
�p�j;t�1�v
dj +
Z �
0
�(1 + rf )p�j;t
�1�vdj
� 11�v
: (32)
When the country is in a default state, �nal goods �rms cannot raise the working cap-
ital necessary to import intermediate goods. Thus, the price of aggregated imported
intermediate goods in this state is:
p�t =
�Z 1
�
�p�j;t�1�v
dj
� 11�v
: (33)
A.3 Domestic Intermediate Goods Firms
A representative domestic intermediate goods �rm produces intermediate goods from
labor Lmt and sells them to �nal goods �rms. The technology of domestic intermediate
goods production is:
mdt = AI (Lmt )
; (34)
where both parameters of AI and 2 [0; 1] are the time-invariant state of TFP and
the labor share respectively in intermediate goods production. The pro�t maximization
problem of intermediate goods �rms is:
maxLmt
�mt = pmt AI (Lmt )
� wtLmt : (35)
Then, the �rst-order condition with respect to Lmt is:
wt = pmt AI (Lmt )
�1 : (36)
A.4 Competitive Equilibrium
We de�ne the competitive equilibria of both the inability-to-repay default model and
the strategic default model. For convenience, we summarize state vectors as sI 2 fA, g,B, �, b�g for the former model and sS 2 fA, g, B, �g for the latter model. The prime
28
symbol represents variables at the next period (i.e., B0 represents the debt outstanding
at the next period.).
De�nition 1
A recursive equilibrium of the inability-to-repay default model is de�ned as, given the
vector of state variables sI , the �scal limit in the next period b�0 and tax rate �(sI), a
set of government policies fB0(sI), G(g), T (sI)g, private allocation fC(sI), Y (sI), L(sI),M(sI), md(sI), m�(sI), Lf (sI), Lm(sI)g, factor prices fw(sI), p�(sI), pm(sI)g, and theprice of government bonds q(A, g, B0, �n, b
�0) such that:
(a) Government policies satisfy the rules of government expenditure and taxation (1),
and for � = �n, given q(A, g, B0, �n, b
�0), the amount of newly issued government bonds
follows the budget constraint (2); (b) For � = �n, given the default set �I(B; �n; b
�) and
the probability of default P e(A, g, B0, �n, b�0), the price of government bonds q(A, g,
B0, �n, b�0) satis�es the foreign investors�zero pro�t condition (3); (c) Households, �nal
goods �rms, and intermediate goods �rms solve their optimization problems respectively;
(d) The market for domestic intermediate goods clears; (e) The market for labor clears,
Lt = Lft + Lmt ; (f) The market for �nal goods clears, Yt � Tt � p�tm�t = Ct.
De�nition 2
A recursive equilibrium of the strategic default model is de�ned as, given the vector of
state variables sS, tax rate �(sS) and value functions V (sS), Vn(sS) and Vd(sS), a set
of government policies fd(sS), B0(sS), G(g), T (sS)g; private allocation fC(sS), Y (sS),L(sS), M(sS), md(sS), m�(sS), Lf (sS), Lm(sS)g, factor prices fw(sS), p�(sS), pm(sS)g,and the price of government bond q(A, g, B0, �n) such that:
(a) Government policies satisfy the rules of government expenditure and taxation (1),
for � = �n, given q(A, g, B0, �n), the amount of newly issued government bonds follows
the budget constraint (2) and the government solves its optimization problem (16); (b)
For � = �n, given the default set �S(B, �n) and the probability of default P
e(A, g,
B0, �n), the price of government bonds q(A, g, B0, �n) satis�es the foreign investors�
zero-pro�t condition (3); (c) Households, �nal goods �rms, and intermediate goods �rms
solve their optimization problems respectively; (d) The markets for domestic intermediate
goods, labor and �nal goods clear.
29
In terms of the market clearing condition of �nal goods, the aggregate constraints
are di¤erent in the normal and default states as Yt�p�tm�t � (Bt� qtBt+1) = C t+G t and
Yt�p�tm�t +�
Ft = Ct+Gt respectively. However, by substituting the government�s budget
constraints in the normal state (2) and that in the default state, Tt + �Ft = Gt, for each
state of aggregate constraint, the same market clearing condition is derived regardless of
the government state.
30
Appendix B. Computational Algorithm
B.1 Expected Surplus and Fiscal Limit
1. Discretize the state space of tax rate (�), and TFP (A) and government expenditure
normalized by GDP (g) in terms of deviation from the respective steady-state value.
Take 101 grid points for the tax rate uniformly in the interval between 0.15 and
0.60.31 Tauchen�s method (1986) is applied to obtain the state space of TFP and
government expenditure, taking 25 grid points uniformly by setting the center points
to zero.
2. Calculate endogenous variables (Ct; Yt; Lt;mdt ;m
�t ; L
ft ; L
mt ; Tt; Gt; wt; p
mt ; p
�t ) for all
grid points.
3. Calculate the expected surplus and �scal limit separately.
(a) Expected Surplus: Aggregate the expected future �scal surplus discounted by
the risk-free interest rate EtP200
i=0
�1
1+rf
�t+i(Tt+i�Gt+i) given the correspond-
ing tax rate on each grid point.
(b) Fiscal Limit: Find the tax rate that maximizes the tax revenue in each state
variable (At; gt), and subtract the political disturbance. Then, draw the fu-
ture shocks for (At+i; gt+i) given the initial state (At; gt) and aggregate the
discounted future maximum �scal surplus over a 200-quarter horizon. Repeat
this procedure 5,000 times for all initial states of (At; gt). Finally, calculate
the cumulative distribution function of default probability.
B.2 Discrete State Space of Inability-to-Repay Default
1. Discretize the state space of TFP (At), government expenditure (gt) and govern-
ment debt (Bt), taking the same number of grid points for TFP and government
expenditure described in the previous section and 101 grid points for the govern-
ment debt Bt in the interval between -0.5 and 3.0 times the steady-state level of
GDP. Then, make an initial guess for the issuance of government bonds f b0 .
31We only study the tax rates within this range. This is because if the tax rate is too high, the�rst-order conditions of �nal goods �rms will not be obtained due to negative pro�ts as �rms produceat least the amount of government expenditure. Also, if the tax rate is too low, the �scal limit will benegative because of low tax revenues.
31
2. Given the tax rate rules, � lrt , �est and �
mat , evaluate the probability of default P
I;e
with new government bond f b0 using a piecewise linear interpolation, and compute
other endogenous variables.
3. Update the guess for government bond issuance f b1 from the government intratem-
poral budget constraint (2) and the pricing equation of government bond (3).
4. Accept the decision rule f b1 if the di¤erence between the updated and old decision
rules f b0 is small enough (i.e. sup f b1 � f b0
< �).32 Otherwise, go back to step 2
with the updated decision rule f b0 = f b1 .
5. Finally, select the tax rule � t from the state spaces for (� lrt , �est and �
mat ) that satis�es
the rule (12) and the corresponding decision rule for government bond issuance f b
and endogenous variables.
B.3 Discrete State Space of Strategic Default
1. Discretize the state space of TFP (At), government expenditure (gt) and tax rate
(� t). Set the tax rate (� t) from its linear rule (4), corresponding to the debt out-
standing that ranges from -0.5 to 3 times the steady-state level of GDP. Take the
same number of grid points for TFP, government expenditure and tax rate as the
prior sections.
2. Make initial guesses for the non-default and default value functions Vn;0 and Vd;0
respectively, and initialize the government bond price q0 to be 1=(1 + rf ).
3. The government�s value function iteration.
(a) Given q0, calculate the private allocations and utilities for each state space in
non-default and default. Then, derive the tax rate in the next period and the
corresponding expected value functions using a piecewise linear interpolation.
(b) Calculate the updated value functions Vn;1 and Vd;1 from the expected value
functions and utilities, following equations (17) and (18) respectively.
(c) If the di¤erences between current and updated government value functions are
small enough (i.e. supfjVn;1 � Vn;0j, jVd;1 � Vd;0jg < "g), stop the iteration.Otherwise, go back to step (a) with the updated government value functions.
32We checked the local uniqueness of the solution by perturbing the policy function f b, and con�rmedits convergence.
32
4. Calculate the default set �S and the probability of default P S;et , using a piecewise
linear interpolation on the tax rate grid points. Then, derive the price of government
bond q1.
5. If the di¤erence between the current and updated government bond price is small
enough (i.e. supjjq1 � q0jj < "g), stop the iteration. Otherwise, go back to step 3with the updated price of the government bond.
33
Appendix C. Quantitative Analysis under Di¤erent
Values of Government Expenditure
In section 4, we explained how the current level of TFP a¤ects the probability of default
under both the inability-to-repay default model and the strategic default model. In
this Appendix, we examine how the other state variable, i.e., government expenditure,
in�uences the results. Figure C shows these two cases under di¤erent values of government
expenditure, keeping the value of TFP at the steady-state value.
Figure C: Comparison of the Two Sovereign Default Models (Greece)
(a) Probability ofInability-to-Repay Default (b) Value Function of Strategic Default
0 0.5 1 1.5 2 2.5 3Debt / GDP
0
0.2
0.4
0.6
0.8
1
High gSteady gLow g
0 0.5 1 1.5 2 2.5 3Debt / GDP
0
0.2
0.4
0.6
0.8
1V
nV
d
Notes 1: Debt-to-GDP is measured relative to the steady-state GDP.
2: "High g" and "Low g" represent government expenditure at a level 5 percent higher and
lower than the steady state respectively.
3: TFP is �xed at the steady state.
Panel (a) shows that current high (low) government expenditure increases (decreases)
the probability of inability-to-repay default. However, similar to the TFP, variations in
the probability are small because government expenditure is assumed to return to the
steady-state level. Panel (b) shows that the non-default value (Vn) is larger than the
default value (Vd) in the plausible region of debt-to-GDP ratio, indicating that without
changes in TFP level, changes in government expenditure on their own would not have
triggered the Greek default.
34
Appendix D. Counterfactual Analysis
In the current paper we argue that the Greek default is likely attributable to the gov-
ernment�s inability to repay its debt as suggested by our simulation analysis. However,
why does the strategic default model fail to explain the Greek default? To see this more
clearly, in this appendix, we explore an extreme counterfactual scenario under which
strategic default can emerge as a plausible scenario. To be speci�c, we examine the case
where the tax rate under the default state is reduced by 8 percent from the baseline simu-
lation, equivalent to political disturbance, and government expenditure is 5 percent lower
than the baseline during the default state. Clearly, under these premises, the economic
contraction after the default would become more moderate, and therefore the government
has a higher incentive to choose to default.
Figure D: Trigger of Default
(a) Value Function of Strategic Default (b) Default Region
0 0.5 1 1.5 2 2.5 3Debt / GDP
1.5
1
0.5
0
0.5
1
Vn
Vd
High AMiddle ALow A
7.5
5.6
3.7
1.9
0
1.9
3.8
5.6
7.5
TFP
0 0.5 1 1.5 2 2.5 3Debt / GDP
NonDefault
Incapabilitytorepay
Default inBoth Ways
StrategicDefault
DefaultedPoint
Notes 1: Debt-to-GDP is measured relative to the steady-state GDP.
2: TFP in panel (b) represents the deviation from the steady state.
3: The tax rate and government expenditure during default are assumed to be lower for 8% points
and 5% points than the baseline respectively.
Panel (a) in Figure D shows the di¤erence between the value function for non-default
and that for default. The default value exceeds the non-default value at the point where
the debt-to-GDP ratio is around 160 percent when TFP is at the steady-state level. The
threshold is lower when TFP is lower. Panel (b) shows the default regions with respect
to the TFP for di¤erent values of debt-to-GDP ratio. Again, we �xed the government
expenditure at 5 percent lower than the steady-state value. In this case, the strategic
35
default region emerges when the debt-to-GDP ratio is relatively high. For example,
provided that TFP is at the steady-state value, the government chooses to default as a
result of the strategic decision when the debt-to-GDP ratio exceeds 160 percent. When
the ratio exceeds 180 percent, the government can default either by its inability to repay
its debt or by strategic decision. These two panels show that strategic default can occur
only under the assumptions of a moderate economic downturn during the default state,
which is, however, at odds with observations in the case of the Greek default.
36
Appendix E. Calibration
In this appendix, we explain calibrated parameter values for the Argentinean economy.
Most parameters are set following Mendoza and Yue (2012). The risk-free interest rate rf
is set at 2 percent, based on the average money market rate between 1993Q4 and 2001Q3.
The households� subjective discount factor � is set at 0.98. The ratio of government
expenditure to GDP is set at 21.9 percent, based on the historical average between
1993Q4 and 2001Q3. The steady-state tax rate � and the government expenditure-to-
GDP G are set as equal. The government debt-to-GDP B is set at 34 percent, based
on the historical average of the corresponding series between 1993Q4 and 2001Q3. The
political disturbance % is set at 0.33, in accordance with the International Country Risk
Guide�s (ICRG�s) index. We summarize the calibrated parameters for the Argentinean
economy in Table E. Other parameters not listed in this table are set at the same values
used in the Greek case.
Table E: Calibration to the Argentinean Economy
Parameter Value Target/Source�: Households�discount factor 0.98 1�rf�M : Intermediate input share in �nal goods production 0.43 Mendoza and Yue (2012)�L: Labor input share in �nal goods production 0.40 Mendoza and Yue (2012) : Substitution elasticity across intermediate goods 2.86 Mendoza and Yue (2012)�: Substitution elasticity within intermediate goods 2.44 Mendoza and Yue (2012)�: Armington weight of domestic inputs 0.62 Mendoza and Yue (2012)rf : Risk-free interest rate 0.02 Money Market RateG: Steady-state government expenditure 0.0106 IADB (G=GDP = 0:219)B: Steady-state government debt 0.0163 IMF (B=GDP = 0:335)� : Steady-state tax rate 0.219 Steady-state gov�t expenditure%: Political disturbance 0.33 Bi, Shen and Yang (2014)
37
Appendix F. Data
Table F: Data
Variables Term Frequency Sources<Greece>Public debt-to-GDP 1999-2016 quarterly OECDGovernment expenditure-to-GDP 1999-2008 quarterly (SA) IMFGovernment tax revenue-to-GDP 1999-2016 quarterly (SA) IMFGDP 1999-2016 quarterly (SA) OECDConsumption 1999-2016 quarterly (SA) OECDImported intermediate goods early 2000s - OECD*Imports of goods and services 1999-2016 quarterly (SA) OECDDomestic intermediate goods 1999-2016 annual ECBLabor 1999-2016 quarterly (SA) OECDYields on Greek and German gov�t bonds 1999-2016 quarterly OECDTFP 1999-2016 quarterly (SA) OECD
<Argentina>Public debt-to-GDP 1993-2001 quarterly IMFGovernment expenditure-to-GDP 1993-2001 quarterly (SA) IADBMoney market rate 1993-2001 quarterly IMF
<World>Default Episodes 1982-2013 annual SUGDP 1979-2016 annual WB
Notes 1: SA indicates that data is seasonally adjusted.
2: * represents the OECD�s STAN Input-Output Data.
3: IADB represents the Inter-American Development Bank.
4: SU represents Schmitt-Grohe and Uribe (2017).
Most of the Greek data we use is that of from 1999Q1 to 2016Q4, except for gov-
ernment expenditure-to-GDP and imported intermediate goods, where the data for gov-
ernment expenditure-to-GDP is that from 1999Q1 to 2008Q4, since the steady state and
exogenous shock associated with government expenditure-to-GDP needs to be calibrated
with pre-crisis data, and the data for imported intermediate goods is that from the early
2000s due to data limitation. Most Argentinean data we use is from 1993Q4 to 2001Q3
which is 1 quarter prior to the crisis.
Note also that government expenditure is the government�s total expenditure, includ-
ing social bene�ts minus interest payments; tax revenue includes social contributions;
and labor is total employment for those aged 15 and over. Yields on both Greek and
38
German government bonds are measured from 10-year bond yields. TFP is de�ned as
GDP per person employed. The world default episodes are obtained from Schmitt-Grohe
and Uribe (2017). The GDP data for defaulting countries is from World Development
Indicators, published by the World Bank (WB).
39